PAMM · Proc. Appl. Math. Mech. 8, 10969 – 10970 (2008) / DOI 10.1002/pamm.200810969

Adaptive multiresolution schemes for reaction-diffusion systems

Mostafa Bendahmane ∗1, Raimund Burger ∗∗1, Ricardo Ruiz ∗∗∗1, and Kai Schneider †2

1 Departamento de Ingenierıa Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile.2 Centre de Mathematiques et d’Informatique, Universite de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France.

This note outlines a fully adaptive multiresolution scheme with local time stepping for the efficient numerical solution of (pos-sibly degenerate) reaction-diffusion systems. A new numerical example showing Turing-type pattern formation is presented.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In [1] we present a fully adaptive multiresolution (MR) scheme for spatially 2D, possibly degenerate reaction-diffusion sys-tems, focusing on models of combustion, pattern formation, and chemotaxis. Solutions of these equations in these applicationsexhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computa-tional effort in zones of strong variation. The MR scheme is based on finite volume discretizations with explicit time stepping.The MR representation of the solution is stored in a graded tree (“quadtree”), whose leaves are the non-uniform finite volumeson the borders of which the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leavesthat are smaller than a threshold value, substantial data compression and CPU time reduction is attained. Our version of theMR method includes a locally varying adaptive time stepping strategy similar to that by Muller and Stiriba [5]. Numericalexperiments presented in [1] illustrate the effectiveness of the adaptive MR method. It turns out that local time steppingaccelerates the adaptive MR method by a factor of two, while the error remains controlled.

2 Reaction-diffusion system, multiresolution scheme and numerical example

We consider the domain QT := Ω× (0, T ), Ω := (0, 1)2, its boundary ΣT := ∂Ω× (0, T ), and the reaction-diffusion system

ut = γf(u, v) + ∆u, vt = γg(u, v) + d∆v on QT , (1)

u(x, 0) = u0(x), v(x, 0) = v0(x) on Ω, ∇u · n = ∇v · n = 0 on ΣT . (2)

In [1], this system is studied with more general degenerate diffusion terms. This system models several phenomena includingcombustion [1, 7], but is considered here as a well-known model of pattern formation in mathematical biology. Under anumber of structural conditions relating the functions f(u, v) and g(u, v) and their derivatives to the parameters γ and d, thesystem produces stationary solutions with Turing-type spatial patterns (see [6] for details). To produce this effect, we selectthe kinetics f(u, v) = a− u + u2v and g(u, v) = b− u2v, with the parameters a = −0.5, b = 1.9, d = 4.8, and γ = 800 [6].

We employ a standard finite volume scheme to discretize, which is described here for a uniform grid. The domain Ω ispartitioned into control volumes (Ωij)(i,j)∈Λ, where Λ is an index set, defining Ωij := [xi−1/2, xi+1/2] × [yj−1/2, yj+1/2],∆x := xi+1/2 − xi−1/2, ∆y := yj+1/2 − yj−1/2, for all (i, j) ∈ Λ, and ∆x := min∆x, ∆y. Let qij(t) denote the cellaverage over Ωij of a quantity q at time t. The FV scheme is described here for the first equation of (1); for the secondequation, we replace u by v and f(u, v) by g(u, v). Integrating the respective equation and averaging over Ωij yields thefolowing expression, where D denotes the RHS of the PDE except for the reaction term:

1|Ωij |

∫∫Ωij

ut(x, t) dx =1

|Ωij |∫∫

Ωij

D(u(x, t),∇u(x, t)

)dx +

1|Ωij |

∫∫Ωij

f(u(x, t), v(x, t)

)dx.

We discretize D via Dij := − 1∆x (Fi+1/2,j − Fi−1/2,j) − 1

∆y (Fi,j+1/2 − Fi,j−1/2) defining Fi+1/2,j := − 1∆x (ui+1,j − uij)

and Fi,j+1/2 := − 1∆y (ui,j+1 − uij). The reaction term is approximated by fij ≈ f(uij , vij). A first-order Euler time dis-

cretization yields the formula un+1ij = un

ij + ∆tγfij + ∆tDij(S(unij), ∆x), vn+1

ij = vnij + ∆tγgij + d∆tDij(S(vn

ij), ∆x),where S(·) denotes the stencil used for computing Dij . This scheme is stable under the CFL condition

λγ(‖fu‖∞ + ‖fv‖∞ + ‖gu‖∞ + ‖gv‖∞) + 8µd ≤ 1, λ := ∆t/∆x, µ := ∆t/∆x2. (3)

The adaptive MR method is an efficient space-time adaptive implementation of the FV scheme, which relies on storingthe numerical solution on a sequence of nested dyadic grids defined on Ω. Basically, we define a finest grid on Ω with the

∗ e-mail: mostafab@ing-mat.udec.cl. MB is supported by Fondecyt project 1070682.∗∗ Corresponding author. e-mail: rburger@ing-mat.udec.cl. RB is supported by Fondecyt project 1050728 and Fondap project 15000001.∗∗∗ e-mail: rruiz@ing-mat.udec.cl† e-mail: kschneid@cmi.univ-mrs.fr

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10970 Sessions of Short Communications 23: Scientific Computing

0 0.5 1x 0 0.5 1x 0 0.5 1x 0 0.5 1x0

0.5

1

y

Fig. 1 Numerical simulation of Turing-type pattern formation. From left to right: component u at t = 0.05, adaptive mesh at t = 0.05,component u at t = 1.5 and adaptive mesh at t = 1.5. The solution assumes values 0.8 < u < 1.7 and 0.1 < u < 2.1 for t = 0.05 andt = 1.5, respectively. The adaptive mesh is represented by the centers of the finite volumes.

t V η L1−error in (u,v) L2−error in (u,v) L∞−error in (u,v)

0.05 7.94 12.2072 (5.83 × 10−4, 4.02 × 10−4) (3.89 × 10−4, 2.46 × 10−4) (1.95 × 10−3, 9.08 × 10−4)0.5 10.63 11.7857 (5.91 × 10−4, 4.77 × 10−4) (4.97 × 10−4, 4.31 × 10−4) (9.59 × 10−4, 8.44 × 10−4)1.5 12.23 11.9140 (7.62 × 10−4, 5.75 × 10−4) (8.08 × 10−5, 3.32 × 10−4) (1.22 × 10−3, 1.16 × 10−3)

Table 1 Turing model: Corresponding simulated time, speedup V , data compression rate η, and normalized errors.

meshwidth ∆x = ∆y = ∆x = 2−L, L ∈ N. We may represent the cell averages of a function defined on the finest grid bystoring the average of four neighbouring cells that form a cell of the next coarsest grid, plus three independent differences,called details. Repeating this process successively for all coarser levels eventually leads to the multiresolution representationof the solution, which consists of one average value for each cell of the coarsest grid plus a sequence of details corresponding tofiner levels. For a sufficiently smooth function, the details diminish when we proceed from coarser to finer levels of resolution.The decisive benefit of the MR representation accrues from an operation called thresholding, that is, we set to zero all detailsthat are smaller in absolute value than a tolerance εl = 22(l−L)εR, where εR is a reference tolerance and l = 0, 1, . . . , L arethe levels of multiresolution (l = L is the finest). Roughly speaking, the numerical solution in the graded tree is re-organizedafter each time iteration, and the additional error introduced by thresholding will not deteriorate the rate of convergence of theFV scheme if εR is chosen in a particular way that is compatible with the CFL condition (3), see [4] for a rigorous analysisfor the case of conservation laws and [1, 2, 3] for further details on our implementation. Clearly, this procedure naturallysuggests that we employ a graded tree data structure for the representation of the solution. In this so-called quadtree (i.e., anode will have up to four sons), the leaves correspond to the finest level of resolution and form the finite volumes on whichthe numerical discretization is eventually applied.

The performance of the MR device is measured by the speedup V and the data compression rate η, which are defined by

V =CPU time of FV scheme on finest grid

CPU time of MR version of FV scheme, η =

#cells of finest grid#cells of finest grid · 2−2L + #leaves .

An additional speedup of a factor of roughly two is attained by local time stepping (first proposed by Muller and Stiriba [5]).The basic idea is to advance parts of the solution corresponding to leaves lying on coarser levels of resolution by larger timesteps than those corresponding to fine levels, and to ensure stability by strictly enforcing the CFL condition. This requires acareful synchronization after each coarse-level time iteration; this task is, however, greatly facilitated by the graded nature ofthe tree. See [1] for further technical details that have been omitted in this note, and which also include the precise evaluationof numerical fluxes and the creation of a safety zone in the graded tree.

Our example (Figure 1) starts from a random perturbation of the steady state (u0 = a + b = 1.4, v0 = b/(a + b)2 =0.96939). We use L = 8 and εR = 7.82 × 10−4. The effectiveness of the MR method is illustrated in Table 1.

References

[1] M. Bendahmane, R. Burger, R. Ruiz, K. Schneider, Preprint 2007-35, DIM, U. de Concepcion; submitted.[2] R. Burger, R. Ruiz, K. Schneider, M. Sepulveda, J. Engrg. Math. 60 (2008) 365–385.[3] R. Burger, R. Ruiz, K. Schneider, M. Sepulveda, M2AN Math. Model. Numer. Anal., to appear.[4] A. Cohen, S. Kaber, S. Muller, M. Postel, Math. Comp. 72 (2002) 183–225.[5] S. Muller, Y. Stiriba, J. Sci. Comput. 30 (2007) 493–531.[6] J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third Edition, Springer-Verlag, New York, 2003.[7] O. Roussel, K. Schneider, Combust. Theory Modelling 10 (2006) 273–288.

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